where, \(\hat{P}_n\)  is the linear combination of \(n\) -step Wilson loops in \(W(n)\)  and is normalized to one when \(U_{x, \mu }= \mathbf {1}\) .
When we view this shift of the gauge action as a shift in improvement parameters in the parameter space of improved gauge actions, we find that, at least in these low-order terms, the magnitude of the shift is much smaller than those for typical improved actions:
For example, in the Iwasaki improved gauge action, \(S_g = -6N_{\rm site} \beta (c_0 \hat{P} + 2c_1 \hat{R})\)  with \(\hat{R}\)  the \(1 \times 2\)  Wilson loop, the improvement parameters are \(c_0 = 3.648\)  and \(c_1 = -0.331\) , i.e., the ratio of the absolute values of the 4-step and 6-step terms is about \(5.5 : 1\)  [1]}.
On the other hand, the ratio of the 4-step term to the 6-step term in Eq. (REF ) is about \(4000 : N_{\rm f}\)  even at \(\kappa \sim 0.1\) .
The magnitude of coupling parameters is similar also in other improved gauge actions [2]}, [3]}, [4]}.
Therefore, the shift in gauge coupling parameters due to the dynamical quark effect is quite small at \(\kappa \mathrel {\unknown. {<}}{\sim }\) 0.1\(.\footnote {We also note that, since the sign of the 6-step loop term of typical improved actions is negative while that of the W (6) term is positive, a slight unimprovement is required to reproduce the dynamic quark effect.}Because a slight shift in improvement parameters mainly affects the lattice discretization errors, the Wilson loop terms will not affect characteristic features of the system in the continuum limit,though the convergence of the hopping parameter expansion may rapidly worsen close to the chiral limit.In contrast, the Polyakov-type loop terms affect like external magnetic fields in spin models, and thus can change the nature of the phase transition.\)
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